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Distributionally Robust Optimization via Diffusion Ambiguity Modeling

Jiaqi Wen, Jianyi Yang

TL;DR

This work introduces a diffusion-based ambiguity set for distributionally robust optimization (DRO) to address distribution shifts while preserving consistency with the nominal data. By parameterizing distributions through diffusion models and constraining their score-matching loss, the authors obtain a tractable inner maximization within a finite space, solved via dual learning (mu) and policy-gradient or PPO methods. They establish stationary convergence guarantees for the proposed D-DRO algorithm, including bounds on inner-maximization error and KL divergence to the nominal distribution, supported by Moreau-envelope-based interpretations. Empirically, D-DRO yields superior out-of-distribution generalization on renewable energy forecasting tasks, outperforming ML, diffusion-augmented ML, and conventional DRO baselines across diverse noisy and nonstationary conditions.

Abstract

This paper studies Distributionally Robust Optimization (DRO), a fundamental framework for enhancing the robustness and generalization of statistical learning and optimization. An effective ambiguity set for DRO must involve distributions that remain consistent with the nominal distribution while being diverse enough to account for a variety of potential scenarios. Moreover, it should lead to tractable DRO solutions. To this end, we propose a diffusion-based ambiguity set design that captures various adversarial distributions beyond the nominal support space while maintaining consistency with the nominal distribution. Building on this ambiguity modeling, we propose Diffusion-based DRO (D-DRO), a tractable DRO algorithm that solves the inner maximization over the parameterized diffusion model space. We formally establish the stationary convergence performance of D-DRO and empirically demonstrate its superior Out-of-Distribution (OOD) generalization performance in a ML prediction task.

Distributionally Robust Optimization via Diffusion Ambiguity Modeling

TL;DR

This work introduces a diffusion-based ambiguity set for distributionally robust optimization (DRO) to address distribution shifts while preserving consistency with the nominal data. By parameterizing distributions through diffusion models and constraining their score-matching loss, the authors obtain a tractable inner maximization within a finite space, solved via dual learning (mu) and policy-gradient or PPO methods. They establish stationary convergence guarantees for the proposed D-DRO algorithm, including bounds on inner-maximization error and KL divergence to the nominal distribution, supported by Moreau-envelope-based interpretations. Empirically, D-DRO yields superior out-of-distribution generalization on renewable energy forecasting tasks, outperforming ML, diffusion-augmented ML, and conventional DRO baselines across diverse noisy and nonstationary conditions.

Abstract

This paper studies Distributionally Robust Optimization (DRO), a fundamental framework for enhancing the robustness and generalization of statistical learning and optimization. An effective ambiguity set for DRO must involve distributions that remain consistent with the nominal distribution while being diverse enough to account for a variety of potential scenarios. Moreover, it should lead to tractable DRO solutions. To this end, we propose a diffusion-based ambiguity set design that captures various adversarial distributions beyond the nominal support space while maintaining consistency with the nominal distribution. Building on this ambiguity modeling, we propose Diffusion-based DRO (D-DRO), a tractable DRO algorithm that solves the inner maximization over the parameterized diffusion model space. We formally establish the stationary convergence performance of D-DRO and empirically demonstrate its superior Out-of-Distribution (OOD) generalization performance in a ML prediction task.
Paper Structure (37 sections, 5 theorems, 67 equations, 4 figures, 5 tables, 1 algorithm)

This paper contains 37 sections, 5 theorems, 67 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. DRO and Ambiguity Modeling
  3. Method
  4. Diffusion Ambiguity Modeling
  5. Diffusion-Based Inner Maximization
  6. D-DRO Algorithm
  7. Analysis
  8. Experiment
  9. Conclusion
  10. Related Work
  11. Distributionally Robust Optimization
  12. Generative Models for Robust Learning
  13. Preliminaries of Diffusion Models
  14. Details of InnerMax in D-DRO Algorithm \ref{['DAG-DRL']}
  15. Objective Derivation by Policy Gradient
  16. ...and 22 more sections

Key Result

Theorem 1

Let $\theta^*$ be the optimal diffusion parameter that solves the inner maximization eqn:objective given a variable $w$. If the expected score-matching loss is bounded as $J(\theta)\leq \bar{J}$ and the step size is chosen as $\eta\sim\mathcal{O}(\frac{1}{\sqrt{K}})$, the inner maximization error ho where the outer expectation is taken over the randomness of output selection. In addition, given th

Figures (4)

  • Figure 1: Gaussian perturbation strength
  • Figure 2: Perlin perturbation strength
  • Figure 3: Cutout perturbation strength
  • Figure 4: Effect of budget $\epsilon$ in D-DRO

Theorems & Definitions (5)

  • Theorem 1: Convergence of Inner Maximization
  • Theorem 2: Convergence of D-DRO
  • Lemma 3
  • Lemma 4
  • Lemma 5